Bicrossproduct Structure of the Quantum Weyl Group

Abstract

We show that the affine quantum group U q (ŝl 2 ) is isomorphic to a bicrossproduct central extension CZ χ ◮⊳U q (Lsl 2 ) of the quantum loop group U q (Lsl 2 ) by a quantum cocycle χ, which we construct. We prove the same result for U q (ĝ) in R-matrix form.Keywords: affine quantum group -central extension -quantum cocycle -loop group -R-matrix.The required theory of (non-Abelian) Hopf algebra extensions was already introduced (by the author) in [4], where it is shown that extensions generally have the form of a cocycle bicrossproduct. The quantum Weyl group is already known to be of this form [5]. In general however, it can be hard to come up with the cocycle data required in the construction. Our first result, in Section 2, provides a general solution to the contruction problem in the case of a central extension.We give the case of U q (ŝl 2 ) in detail, in Section 3. In Section 4 we give the result more generally using the R-matrix formalism with generators l ± (z) in [6] [2].

Cross Product Bialgebras Part II

“…In the second row all descendants with one trivial morphism µ l , µ r , σ, ρ, ν l , or ν r are listed, in the third row all special types with two trivial morphisms are listed, etc. The cross product bialgebra constructions from [24,18,22,20,5] are descendants of the special cases which are the co-cycle free cross product bialgebras of [5] and the braided versions of bicrossed product bialgebras [20,22] respectively 8 . In the following tables we describe the different types of cocycle cross product bialgebras in more detail; we omit the description of the various cross product bialgebras studied in [24,18,22,20,5].…”

“…Eventually we apply our construction and investigate strong cross product bialgebras according to their (co-)modular co-cyclic structure. In particular we recover all known constructions [24,18,22,20,5] and find various new types of cross product bialgebras. All of them are special versions of the most general strong cross product bialgebra construction.…”

“…We found a universal theory of strong cross product bialgebras with an equivalent (co-)modular co-cyclic characterization in terms of strong Hopf data. The theory unites all known cross product bialgebras [24,18,22,20,21,5] in a single construction. Furthermore various new types of cross product bialgebras arise out of the most general construction.…”

“…This is not an accidential fact but has to do with the subsequent definition of cross product bialgebras where we use precisely these algebra and coalgebra structures. Only then are we able to get all the known cross product bialgebras [24,18,22,20,5]. This kind of symmetry between cross product algebra and cross product coalgebra will be called π-symmetry henceforth.…”

“…Despite those common properties of the cross product bialgebra constructions in [24,18,20,22,30] no theory exists which characterizes all of them as special versions of a single universal construction which in turn is uniquely determined by its (co-)modular co-cyclic structure. A first step towards this objective has been achieved in [5] where we discovered a method to describe cross product bialgebras without co-cycles, generalizing and uniting [18,21,24].…”